Related papers: Exceptional Points for Density Modulo 1

A diffeomorphism f is called super exponential divergent if for every r>1, the lower limit of #Per_n(f)/r^n diverges to infinity as n tends to infinity, where Per_n(f) is the set of all periodic points of f with period n. This property is…

We are concerned with describing the structure of the set of points in the unit interval which, when subjected to rotation by irrational alpha modulo one, for all finite portions of the orbit contain at least as many points in the bottom…

The irrationality exponent of a real number measures how well that number can be approximated by rationals. Real numbers with irrationality exponent strictly greater than $2$ are transcendental numbers, and form a set with rich fractal…

In this paper we produce precise large deviation estimates through the lens of mod-Poisson convergence. We apply a general result to various examples from number theory, Dedekind domains and polynomials over finite fields when an element is…

We consider four related problems. (1) Obtaining dimension estimates for the set of exceptional vantage points for the pinned Falconer distance problem. (2) Nonlinear projection theorems, in the spirit of Kaufman, Bourgain, and Shmerkin.…

We study the dimension theory of limit sets of iterated function systems consisting of a countably infinite number of contractions. Our primary focus is on the intermediate dimensions: a family of dimensions depending on a parameter $\theta…

We study dimensions of sumsets and iterated sumsets and provide natural conditions which guarantee that a set $F \subseteq \mathbb{R}$ satisfies $\overline{\dim}_\text{B} F+F > \overline{\dim}_\text{B} F$ or even $\dim_\text{H} n F \to 1$.…

We study the decomposition of real numbers into sums of L\"uroth sets, which are defined by numbers whose L\"uroth expansions have prescribed digit constraints. We establish several results on the congruence modulo 1 of sums of L\"uroth…

We compute the Parisi overlap distribution for paperfolding sequences. It turns out to be discrete, and to live on the dyadic rationals. Hence it is a pure point measure whose support is the full interval [-1; +1]. The space of paperfolding…

We prove an exponential deviation inequality for the convex hull of a finite sample of i.i.d. random points with a density supported on an arbitrary convex body in $\R^d$, $d\geq 2$. When the density is uniform, our result yields rate…

We study the problem of conjugating a diffeomorphism of the interval to (positive) powers of itself. Although this is always possible for homeomorphisms, the smooth setting is rather interesting. Besides the obvious obstruction given by…

We prove that every closed set which is not sigma-finite with respect to the Hausdorff measure H^{N-1} carries singularities of continuous vector fields in the Euclidean space R^N for the divergence operator. We also show that finite…

The study of the geometry of $n$-uniform measures in $\mathbb{R}^{d}$ has been an important question in many fields of analysis since Preiss' seminal proof of the rectifiability of measures with positive and finite density. The…

By improving upon previous estimates on a problem posed by L. Moser, we prove a conjecture of Erd\H{o}s that the density of any measurable planar set avoiding unit distances cannot exceed $1/4$. Our argument implies the upper bound of…

Complex eigenvalues of random matrices $J=\text{GUE }+ i\gamma \diag (1, 0, \ldots, 0)$ provide the simplest model for studying resonances in wave scattering from a quantum chaotic system via a single open channel. It is known that in the…

Exceptional points are values of the spectral parameter for which the homogeneous Faddeev scattering problem has a non-trivial solution. We study the existence/absence of exceptional points for small perturbations of conductive potentials…

In this article, we study two problems concerning the size of the set of finite point configurations generated by a compact set $E\subset \mathbb{R}^d$. The first problem concerns how the Lebesgue measure or the Hausdorff dimension of the…

Studies have shown that quantum states reside in a Hilbert space bundle. When a quantum system depends on continuous external parameters, these parameters define additional dimensions in the base space of the bundle. While much of the…

We study the asymptotic behavior of the maximum interpoint distance of random points in a $d$-dimensional set with a unique diameter and a smooth boundary at the poles. Instead of investigating only a fixed number of $n$ points as $n$ tends…

A previous article was devoted to an analysis of the symmetry properties of a class of first-order delay ordinary differential systems (DODSs). Here we concentrate on linear DODSs. They have infinite-dimensional Lie point symmetry groups…